The Slope of the Instant

This article records tradition as it has been passed down and reported. Its sources are not yet part of the atlas's verified catalogue.

Ask what the speed of a falling object is at one exact instant, and ordinary language runs into a wall: speed is distance divided by time, and at a single instant no time has passed for anything to divide by. The derivative is the answer mathematics eventually gave to that problem, and it took most of the seventeenth century to arrive at it cleanly. The trick, in outline, is to ask the question over a very short interval instead of an instant, get an answer, then ask what happens to that answer as the interval shrinks toward zero without ever quite reaching it. Newton called the result a fluxion; Leibniz called it a differential and by 1675 had settled on the dy/dx notation still taught today. Both men treated the shrinking interval with a looseness that drew sharp criticism, most famously in Bishop Berkeley's 1734 attack, and efforts to tighten the reasoning continued for a century afterward; it was Cauchy, in the nineteenth century, who built the rigorous basis the calculus had been missing, and Weierstrass's own lectures a few decades later took up the same foundational project. What survived the two centuries of repair is exactly the intuition Newton and Leibniz started with: a derivative is simply the slope of a curve at one single point, made precise.